Table 1 Possible effect heterogeneity scenarios. The left panel is a directed acyclic graph (DAG) where Z, X, Y, U represent the instrument, the exposure, the outcome, and the unmeasured confounders, respectively. M is a variable to be considered. The DAG has different possible scenarios, each of which has different arrow directions (or no arrow, denoted by \(\nleftrightarrow\)) for the X-M and M-Y relationship. Our primary simulation study considers covariates corresponding to scenarios 1, 5, 7, and 9. Scenarios 2, 3, 6, and 8 are considered in an additional simulation study, presented in Supplementary Text
DAG | Scenario | X-M relationship | M-Y relationship | M interpretation | Comments |
|---|---|---|---|---|---|
| 1 | \(X \rightarrow M\) | \(M \rightarrow Y\) | collider and mediator | Possible a modifier |
2 | \(X \rightarrow M\) | \(M \leftarrow Y\) | collider | Not a modifier | |
3 | \(X \leftarrow M\) | \(M \rightarrow Y\) | confounder | Possible a modifier | |
4 | \(X \leftarrow M\) | \(M \leftarrow Y\) | N/A | Ill-defined acylic graph | |
5 | \(X \rightarrow M\) | \(M \nleftrightarrow Y\) | collider | Not a modifier | |
6 | \(X \leftarrow M\) | \(M \nleftrightarrow Y\) | not collider/confounder | Not a modifier | |
7 | \(X \nleftrightarrow M\) | \(M \rightarrow Y\) | not collider/confounder | Possible a modifier | |
8 | \(X \nleftrightarrow M\) | \(M \leftarrow Y\) | collider | Not a modifier | |
9 | \(X \nleftrightarrow M\) | \(M \nleftrightarrow Y\) | not collider/confounder | Not a modifier |
